Measure, Integration & Real Analysis, 2021a

106 Chapter 4 Differentiation
EXERCISES 4A
1 Suppose (X, S, μ) is a measure space and h : X → R is an S-measurable
function. Prove that
μ({x ∈ X : |h(x)| ≥c}) ≤ 1 c p ∫
|h| p dμ
for all positive numbers c and p.
2 Suppose (X, S, μ) is a measure space with μ(X) =1 and h ∈L 1 (μ). Prove
that
({
∫
})
∣
μ x ∈ X : ∣h(x) − h dμ∣ ≥ c ≤ 1 ( ∫ (∫ ) ) 2
c 2 h 2 dμ − h dμ
for all c > 0.
[The result above is called Chebyshev’s inequality; it plays an important role
in probability theory. Pafnuty Chebyshev (1821–1894) was Markov’s thesis
advisor.]
3 Suppose (X, S, μ) is a measure space. Suppose h ∈L 1 (μ) and ‖h‖ 1 > 0.
Prove that there is at most one number c ∈ (0, ∞) such that
μ({x ∈ X : |h(x)| ≥c}) = 1 c ‖h‖ 1.
4 Show that the constant 3 in the Vitali Covering Lemma (4.4) cannot be replaced
by a smaller positive constant.
5 Prove the assertion left as an exercise in the last sentence of the proof of the
Vitali Covering Lemma (4.4).
6 Verify the formula in Example 4.7 for the Hardy–Littlewood maximal function
of χ [0, 1]
.
7 Find a formula for the Hardy–Littlewood maximal function of the characteristic
function of [0, 1] ∪ [2, 3].
8 Find a formula for the Hardy–Littlewood maximal function of the function
h : R → [0, ∞) defined by
{
x if 0 ≤ x ≤ 1,
h(x) =
0 otherwise.
9 Suppose h : R → R is Lebesgue measurable. Prove that
is an open subset of R for every c ∈ R.
{b ∈ R : h ∗ (b) > c}
Measure, Integration & Real Analysis, by Sheldon Axler